In light of the generalized gradient-flow framework of [1]} and the relation to minimizing movement schemes, a formal minimization procedure provides the gradient-flow solution
\(\begin{aligned}\partial _t \hat{\rho }+B^* \hat{\jmath }&=0\\\hat{\jmath }&=(\,\partial _2 \mathcal {R}^*)(\hat{\rho },-B\, \partial _{\rho } \mathcal {F}),\\\end{aligned}\) 
